Harmonics in power system

Electrical and Computer Engineering Student

Colorado State University

Minh Anh Thi Nguyen

E-mail: minhanhnguyen@q.com

Minh.nguyen@colostate.edu

Outline History of Harmonics Linear and non-linear load What are harmonics? What cause harmonics? Harmonic current flow Harmonics components What problem do harmonics create? How do you know if you have a harmonics problem? examples If you have a harmonics problem, what should you do? Why harmonic unknown or untreated in electrical distribution

system? How can we treat harmonics system? Unexpected of harmonics created

History of harmonics Power system have always had harmonics present. In early 1890’s harmonics were associated with distorted current

and voltage waveform shape on transmission system. They didn’t cause a lot of problem in industrial setting or office

building as equipment was less sophisticated. Over the last fifteen years, the proliferation of electronic device has

brought subject up-front and personal. As the number of electronic devices increased, so did the number of

other harmonics creating devices. There has been an explosion of microprocessor based equipment

which are also non-linear loads. Examples include computer systems, variable frequency drives, AC/DC converters, electronic ballasts, X-ray machines, MRI equipment ansentinelerruptible power supplies.

Linear loads Linear loads occur when the impedance is constant; then the current is

proportional to or the same as the voltage A linear element in a power system is a component in which the

current is proportional to the voltage. Typical linear loads are incandescent lights and motors,heaters Linear load, the consumer pays for unused energy due to distortion

voltage.

Non-linear loads Non linear loads occur when the impedance is not constant; then the

current is not proportion or the same as the voltage. The current drawn by non-linear loads is not sinusoidal but is

periodic, meaning that the current wave looks the same from cycle to cycle.

Non-linear loads are low impedance so they use as much as three times the peak current as a resistive load.

Typical non-linear loads are electronic switch power supplies, DC/AC drive and induction furnace/arc furnace.

Non-linear load, the consumer pays more for unused a energy due to both voltage and current distortion.

There are two basically two types of non-linear loads: single phase and three phase.

How big is this problem? Since 1965, the introduction of low cost, high efficiency

semiconductor devices has increased the use of electronic( static) power converters throughout industry in the form of variable speed drives for all type of machinery.

After the 1973 oil embargo and associated rapid increase in energy costs, it has been economical and essential to utilize electronic power converters on large systems, as well as to apply power improvement capacitors to minimize the increased cost of energy. These have also generated significant harmonics in power systems.

In 1980, harmonics were recognized as a major technical issue in the USA. Since then, the national electrical code (NEC) has addressed the requirements for equipment and system performance under influence of harmonics for applications in highly non-linear load installations.

Single phase vs. three phase load harmonics Single phase non-linear loads, like personal computers,

electronics ballast and other electronic equipment, generate odd harmonics (3, 5,7,9.. ect.)

The troublesome harmonics for single phase loads are 3rd and multiple of 3rd ( ie. 9th, 15th, ect.). These harmonics are call “triplens” because A, B,C phase triple are all in the same phase with each other.

They will add rather than cancel on the neutral of a 3 phase 4 wires system. This can over load the neutral if it is not sized to handle this type of load.

3 phase non-linear loads like 3-phase DC driven, 3-phase rectifier, for not generate current triplen harmonics (3,9,15). These type of loads generated primarily 5th and 7th current harmonics and a lesser amount of 11th, 13th and higher order.

Triplen harmonic: odd multiple of the 3rd harmonic (3rd, 9th, 15th, 21).

General definition The general definition of harmonic is any periodic signal (waveform) can be

describe by a series of sine and cosine function, also call Fourier series. Every periodic function of form V(t)=V(t+T) can be developed into Fourier’s

series, which is of form:

1st term of the series (n=1) is called direct wave, and the others harmonics according to their original, so that (n=2) is the second wave.

If the function V(t) is symmetric in the proportion to origin the series contains only odd terms.

V t( ) Vdc

1

n

Vns sin n t Vnc cos n t

the coefficients are obtained as follows

Vdc1

0

2tV t( )

d

Vns1

0

2 tV t( ) sin n t

d

where 2T

T is the fundamental period timeVnc

1

0

2 tV t( ) cos n t

dn is an integer

General definition There are several ways of describing the degree of distortion of a current or voltage. Two

that are frequently used are crest factor and total harmonic distortion THD.– The crest factor of a voltage is equal to the peak value divided by the effective (rms)

value.

• A wave having a crest factor less than 1.4 tend to be flat topped. • A crest factor greater than 1.4 indicates a voltage that tends to be pointy.• In the case of a sinusoidal voltage (which evidently has no distortion) the

crest factor is

– The total harmonic distortion THD of a current or voltage is equal to the effective value of all the harmonics divided by the effective value of the fundamental.

• It is seen that sinusoidal voltages and currents have a THD of zero.

Crestfactorpeakvoltage

effectivevoltage

2 1.414

Total harmonics distortion THDIH

IF

2

n

IF2

IH

Total harmonic distortion voltage THD

VH

VF

Where IH=rms value of all the harmonic components combine IF=rms value of the fundamental component of line current.

What are harmonics? Harmonics are like pot holes. Each harmonic is expressed in terms of its order. Harmonics are currents or voltage with frequencies that are integer multiples of

the fundamental power frequency. Individual harmonic frequency will vary in amplitude and phase angle,

depending on the harmonic source In U.S. the fundamental power frequency is 60Hz. Total harmonic distortion is the contribution of all the harmonic frequency

current to the fundamental. Harmonics result from non-linear loads such as electronics switch power

supplies The characteristic harmonics are based on the number of rectifier (pulse number)

used in a circuit and can be determined by the following equation:– Where: n= an integer (1,2,3,4,5….)

p=number of pulse or rectifiers

Hz5 f 300

Hz7 f 420

Hzf 60

th harmonicsh 5h n p( ) 1

th harmonicsh 7h n p( ) 1

p 6n 1example: using a 6 pulse rectifier:

h n p( ) 1h n p( ) 1

What are causes of harmonics? Harmonics are create by a non-linear loads that draw a current in abrupt pulses

rather than a smooth sinusoidal manner. Electronics switching power supplies/converters Harmonics are created by increased use of non-linear devices such as

uninterruptible power supplies (UPS) systems, solid state variable speed motor drives, rectifiers and personal computers.

Variable speed drives are usually referred to by the number of rectifiers in the system.

Harmonic resonance occurs when the capacitor reactance and the system reactance are equal.

All electronic loads generate positive & negative sequence harmonic currents, single phase electronic loads connected phase neutral in a 3 phase 4wires distribution system, also generate zero sequence harmonic currents

Appliances:

– TV set, air conditioners, washing machines, microwave ovens and vacuum cleaners.

– Fax machines, photocopies, printers and so on.

Harmonic current flow Large harmonic currents will circulate between transformer and

capacitor. These currents will result in greater voltage distortion. When a non-linear load draws current, that current passes through

all of the impedance that is between the load and system source. As a result of the current flow, harmonic voltages are produced by impedance in the system for each harmonic.

These voltages sum and when add to the nominal voltage produce voltage distortion. The magnitude of the voltage distortion depends on the source impedance and the harmonic voltage produced.

If the source impedance is low then the voltage distortion will be low.

If a significant portion of the load become non-linear (harmonic currents increase) and/or when a resonant

condition prevails (system impedance increases) the voltage can increased dramatically.

How do you check for harmonics currents? When non-linear loads are a considerable part of total load in the

facility (more than 20%) there is a chance of a harmonic problem. The amount of current distortion produced by the non-linear loads. Measure the current in the neutral of a 3phase 4 wire system. If the

neutral current is considerably higher than value predicted from the imbalance in the phase currents. There is a good possibility of heavy presence of triplen harmonics.

Other signs of current harmonics include inexplicable higher than normal temperature in the transformer, voltage distortion and high crest factor.

Will harmonic currents affect your power costs? Harmonic currents, generated by single and 3-phase non-linear

electronic loads, will cause significant “penalty” losses throughout the electrical distribution system. – For example, distribution transformers, when supporting 100% THD non-

linear electronic office loads, will produce approximately 3.25 time higher losses than when supporting linear loads.

– Penalty losses, produced by the other elements of the subs system, will typically equal to and often substantially exceed the transformer’s penalty losses.

– Penalty losses result in apparatus overheating higher air conditioning costs and high power costs.

– A reduction in penalty losses will produce a very attractive annual saving.

annual_saving Total_KW_savingdollar

KWh

hrs

day

days

year

Total_KW_savingdollar

KW_demandchage

Months

12 months

Total_PFpenalty_saving

Harmonics components Harmonics are divided into different components by their

properties (ranking number). The component class shows the rotational direction of the phase of

the harmonic with respect to the fundamental frequency. Harmonics are divided into three type of components (positive ,

negative and zero sequence). Zero-sequence harmonics: 3,6,9,12… Positive sequence:1,4,7,10, 13… Negative sequence: 2,5,8,11,14 … In addition harmonics are divided into even and odd harmonics In a normal situation, when the curve form is symmetric in

proportion to origin only odd harmonics are generated. A great number of even harmonics is often a sign of fault or some

other unusual operational situation.

How is such component divided obtain

– If n= 1,4, 7, 10, 13...the phase order (phase angle) is the same as or with the direct wave, or harmonics (phase) rotate in the same direction. positive

– If n = 2, 5, 8, 11... the phase order is different than or with direct waves, or harmonics rotate in a different direction. Negative

– If n = 3, 6, 9, 12... the phase angle in all phases is the same, or harmonics do not rotate. Zero

Positive 6 n 1

negative 6 n 1

Zero 6 n 3

Where n is any integer

Three phase Balance and unbalance Systems Balance in three phase system

– Current in a balance system are shifted 1/3 of a cycle compared to each other.

– Real time phase balancing based on X/R

– Saves energy

– There is no such devices know in markets.

– With no harmonic content, the line current are 120 degree out of phase, cancel each other and result in very little neutral current.

Problems due to unbalance the loading:

– Negative voltage sequence

– Increased current in neutral conductor

– Increased neutral to ground voltage

– Overheating of motors/ insulation breakdown

– Reduced motor efficiency

– Motor bearing failure

– Waste energy/high electric bill

– Waste investment & operating capital

What problem do harmonics create? Large load currents in the neutral wires of a 3 phase system.

– Theoretically the neutral current can be up to the sum of all 3 phases therefore causing overheating of the neutral wires. Since only the phase wires are protected by circuit breakers of fuses, this can result in a potential fire hazard.

Overheating of standard electrical supply transformers which shortens the life of a transformer and will eventually destroy it.

– When a transformer fails, the cost of lost productivity during the emergency repair far exceeds the replacement cost of the transformer itself.

High voltage and current distortion exceeding Poor power factor conditions result in monthly utility penalty fees for major users

(factories, manufacturing and industrial) with a power factor less than 0.9. Resonance that produces over-current surges. In comparison, this is equivalent to

continuous audio feedback through a PA system. This results in destroyed capacitors and their fuses and damaged surge suppressors which will cause an electrical system shutdown

False tripping of branch circuit breakers. harmonics can cause false or spurious operations and trips, damaging or blowing components for no apparent reason.

Waste energy/high electric bill Capacitors: can be affected by heat rise increases due to power loss and reduced life on

the capacitors. If capacitor is tuned to one of the characteristic harmonic such as the 5 th or 7th, over voltage and resonance can cause dielectric failure.

How do you know if you have a harmonic problem? Frequent tripping of circuit breakers & fuse blowing Capacitor failures Overloading of transformer neutrals Telephone interference Severe lamp flicker Excessive equipment heating.

– Increased heating is the result of increased copper and iron losses due to the increased frequencies present.

Electrical interference:

– interference may take the form of loss of data, communication interference. Many electronic devices count on regular sinusoidal voltage waves for detection of peaks and/or zero crossing used in timing circuits.

Over voltage

– An over voltage is a voltage above the normal rated or maximum operating voltage of a device or circuit. Harmonic over voltages are caused by local circuit resonant condition that can overstress equipment insulation. One most common form is the tuning of a circuit due to the addition of a capacitor.

Circuits Is circuit a harmonic circuit? Is circuit is linear or non-linear load circuit?

Does the circuit below a harmonic circuit? why?

circuit1 The output of circuit

**** 11/06/03 08:43:53 *********** Evaluation PSpice (Nov 1999) **************

* U:\EE461\Schematic1.sch

**** FOURIER ANALYSIS TEMPERATURE = 27.000 DEG C

**********************************************************************

FOURIER COMPONENTS OF TRANSIENT RESPONSE V(Vout)

DC COMPONENT = 4.738594E+01

HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED

NO (HZ) COMPONENT COMPONENT (DEG) PHASE (DEG)

1 6.000E+01 9.249E+02 1.000E+00 2.181E-02 0.000E+00

2 1.200E+02 3.512E+00 3.798E-03 -8.847E+01 -8.851E+01

3 1.800E+02 2.086E+01 2.256E-02 -1.779E+02 -1.780E+02

4 2.400E+02 3.405E+00 3.681E-03 -8.570E+01 -8.579E+01

TOTAL HARMONIC DISTORTION = 2.316800E+00 PERCENT

R 10 V 1000

IV

R I 100

P I2

R P 100 103

IrmsI

2 Irms 70.711

Prms Irms2

R Prms 50 103

IH1 Irms( )2

59.32 IH1 38.516

THD38.516

59.34 THD 2.598

Circuit 2 The output of circuit 2

  * U:\EE461\Schematic1.sch  **** FOURIER ANALYSIS TEMPERATURE = 27.000 DEG C***********************************************************************FOURIER COMPONENTS OF TRANSIENT RESPONSE V(Vout) DC COMPONENT = 3.388514E+01

HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONENT COMPONENT (DEG) PHASE (DEG) 1 6.000E+01 5.936E+02 1.000E+00 -2.692E+01 0.000E+00 2 1.200E+02 4.194E+01 7.066E-02 4.173E+01 9.558E+01 3 1.800E+02 3.153E+01 5.311E-02 3.283E+01 1.136E+02 4 2.400E+02 2.487E+01 4.190E-02 2.837E+01 1.361E+02 TOTAL HARMONIC DISTORTION = 9.782316E+00 PERCENT

%THD 9.7THDEH

EF

EH I XL

I 42.996I1000

Z Rl

Z 13.258Z R XL

XL 9.048XL 2 f L

R 4.21Rl 10V 1000Hzf 60HL 0.024

Bipolar amplifier circuit

The example

Bipolar amplifier circuit Pspice results TOTAL POWER DISSIPATION 1.66E-01 WATTS **** 09/24/03 10:35:12 *********** Evaluation PSpice (Nov 1999) ************** * U:\EE461\amplifier.sch **** FOURIER ANALYSIS TEMPERATURE = 27.000 DEG C ****************************************************************************** FOURIER COMPONENTS OF TRANSIENT RESPONSE V(Vout)  DC COMPONENT = -2.893278E-01   HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONENT COMPONENT (DEG) PHASE (DEG)   1 1.000E+04 3.822E+00 1.000E+00 -1.782E+02 0.000E+00 2 2.000E+04 6.956E-01 1.820E-01 9.471E+01 4.511E+02 3 3.000E+04 5.561E-02 1.455E-02 1.148E+01 5.461E+02 4 4.000E+04 4.858E-03 1.271E-03 1.541E+02 8.669E+02 TOTAL HARMONIC DISTORTION = 1.825818E+01 PERCENT

H1 3.822

H2 6.956101

H3 5.561102

H4 4.858103

THDH2

2H3

2 H42

H1100 THD 18.258 %

If you have a harmonics problem, what should you do? How have some engineers dealt with harmonics in their system designs?

– To improve system performance and provide the best possible environment for the non-linear loads, a designer’s options have been limited to over-sizing distribution transformer and ‘shared’ neutral conductors.

– As an alternative, branch circuits have been configured with separate neutral conductor for each phase conductor. In either case, branch circuits have been underutilized and limited in their length as a means of reducing voltage distortion and neutral ground voltage (common mode noise) at the loads.

– As an alternative to over-sizing conventional distribution transformers, many designers have specified K-rated transformers. Unfortunately, a k-rated transformer’s higher harmonics impedance cause increase in voltage distortion.

Ih(pu) is the harmonic current expressed in per unit; and h is the harmonic number.

If you suspect there is a harmonics problem, evaluate the situation:– Compile good date on the symptoms.– Take good measurements of your system at various time during different operating cycles– Invite a consulting engineer or power quality expert from your power supplier to assess your

systems.

K_rate

1

h

Ih pu( )( )2

h2

Reducing harmonics There are many ways to reducing harmonics, ranging from variable frequency

drive designs to the addition of auxiliary equipment. The primary methods used today to reduce harmonics are:– Power System design: harmonics can be reduced by limiting the non-linear load to

30% of the maximum transformer’s capacity. However, with power factor correction capacitors installed, resonating conditions can occur that could potentially limit the percentage of non-linear loads to 15% of transformer’s capacity.

• Determine if a resonant condition on the distribution could occur:

– Isolation transformers: An isolation transformer provides a good solution in many cases. The advantage is the potential to “voltage match” by stepping up or down the system voltage, and by providing a neutral ground reference for nuisance ground faults. This is the best solution when utilizing AC or DC drives that use Silicon controlled rectifiers (SCRs) a bridge rectifiers.

hKVA

sc

KVARh = harmonic order

KVAsc

KVA

Zpu

KAVsc

is available short circuit at point of capcitor bank installation

KVAR = capacitor bank size

Reducing harmonics (const.) Line reactors: more commonly used for size and cost, the line

reactor is the best solution for harmonics reduction when compared to an isolation transformer. AC drives that use diode bridge rectifier front ends are best suited for line reactors. Line reactors (commonly referred to as inductors) are available in standard impedance ranges from 1.5%, 3%, 5% and 7.5%.

Harmonics trap filters: used in applications with a high non-linear ratio to system to eliminate harmonic currents. Filters are tuned to a specific harmonic such as the 5th, 7th, 11th , ect. In addition, harmonic trap filters provide true distortion power factor correction. Filters can be designed for several non-linear loads or for and individual.

Why harmonic unknown or untreated in electrical distribution system?

The electrical distribution system of most sites or facilities was never designed to deal with an abundance of non-linear load.

It’s a problem that has only recently begun to be recognized in the building industry.

Within the last decade, the widespread use of computers and switched-mode power supply (SMPS) equipment is turning modern office buildings, factories and industrial plants into high-tech computer environments.

A building or facility unable to fully support today’s technology and the high tech problems that it brings along with it.

Design a hearing aid or heart pacemaker

Output results The output swing of circuit

AtotalPs 3.01 109

Anew 2.59 109

How can we treat harmonics system? Harmonic treatment can be performed by 2 methods: Filter or

cancellation

– A harmonic filter consists of a capacitor bank and inductor coil.

– The filter is designed or tuned to the predetermined none-linear load and to filter a predetermine harmonic frequency range. Usually this frequency range only accounts for one harmonics frequency. This application is mostly used when specified for a uninterruptible power supplies UPS or variable frequency drive motor in a manufacturing plant.

– Harmonic cancellation is performed with harmonic canceling transformer also know as phase shifting transformers.

– A harmonic canceling transformer is a relatively new power quality product for mitigating harmonic problems in electrical distribution systems.

Why are voltage and current harmonics a problem Current harmonics are a problem because they cause increased

losses in customer and utility power system components. Transformers are especially sensitive to this problem and may need to be de-rated as much as 50% capacity when feeding loads with extremely distorted current waveforms.

In addition, current harmonics can distort the voltage waveform and cause voltage harmonic.

Voltage distortion affects not only sentinel electronic load but also electric motors and capacitor banks.

In electric motors, negative sequence harmonics (i.e. 5th, 11th, 17th). in Capacitor banks, is that the reactance (impedance) of a

capacitor bank decrease as the frequency increases. This cause the bank to act as a sink or trap for higher harmonic currents from the surrounding customer and/or utility system. The effect is increased current, increased heating and dielectric stresses that could lead to capacitor bank failure.

IEEE standard STD 519 In 1981, IEEE guide for harmonic control and reactive compensation of static

power converters, originally established levels of voltage distortion acceptable to the distribution system for individual non-linear loads. With the rising increase usage of industrial non-linear loads, such as variable frequency drives, it became necessary to revise the standard.

On April 12,1993, the IEEE published revised a standard limiting the amplitudes of current harmonics, IEEE recommended practices and requirements for harmonics control in electrical power system.

The establish recommended guidelines for harmonic voltages on the utility distribution systems well as harmonics currents within the industrial distribution system.

IEEE 519 voltage limit

– High voltage system can have up to 2% THD where the cause is an HVDC terminal that will attenuate by the time it is tapped for a user.

IEEE standard 519 IEEE 519 Current limit, low voltage system

– Even harmonics are limited to 25% of the odd harmonic limit above.– Current distortion that result in a DC offset, eg. Half –wave converters are

not allowed.– All power generation equipment is limited to these values of current

distortion, regardless of actual Isc/IL.– where Isc = maximum short circuit current at PCC – IL = maximum demand load current (fundamental frequency component)

at PCC.– n =harmonic order– THD= instantaneous value of harmonic distortion– PCC = At point of metering or point where non-linear load meets linear

loads

What will happen if you do not follow the IEEE Std 519

Electronic equipment is susceptible to miss-operation cause by harmonic distortion data in computer network may be corrupted.

Harmonic current may also contaminate audio and video signal interconnecting shielded cables.

Conventional linear loads can also be affected. This will result in loss of torque, overheating and premature failure.

Harmonics currents, generated by single phase and three phase non-linear electronics loads, will cause additional “ penalty” losses throughout the electrical distribution system.

These losses result in apparatus overheating and premature apparatus failure.

Harmonic currents and voltage are not controlled.

Unexpected harmonics creates Clamp-on ammeter found loads as high as 28 A. Considering the circuits were

rated for 20A and should not exceed 16A continuously) this arrangement was a potentially serious electrical accident waiting to happen.

the engineer who designed the system laid it out using building standard light fixtures. Using 3 lamps per fixtures, one ballast operated the center lamp; the other operated the two outboard lamps.

By providing 2 circuits to each fixtures, the engineer reduced the number of contactors required to control the in/outside lamps separate from the building management system.

The power system was 208V/120V, 3-phase, 4-wire. The owner didn’t have enough fixtures on hand for the entire job. Thinking he was doing the owner a favor, the supplier furnished the additional fixtures with 3 lamps switch-able ballasts, instead of the original 2 ballasts per fixture.

First step in finding the cause of the problem was tabulate the load on each circuit. Of 90 circuits, 15 had overload. 4 were overload because the engineer miscalculated. 11 circuits overload appeared to be

due to the difference between the ballasts

originally specified and the ballasts actually provided.

Unexpected harmonics creates With balance linear loads on two phases, the current on the neutral is no greater

than it would be with just a single phase load. This analysis ruled out the theory that supplying the ballast with 2 circuits from different phases was the problem

The specification for the ballast that were actually supplied revealed they were not electronic. The ballast consisted of a 1&2 lamp ballast, all in the same housing. Both ballasts were conventional magnetic units, but the 1 lamp ballast was a normal (low) power factor ballast. The current and power factor rating for all ballast were:

1-lamp operation= 0.87A at 0.40 PF 2-lamp operation=0.77A at 0.85 PF 3-lamp operation=1.44A at 0.85 PF Part of the problem was the engineer expected the 1-lamp circuit to draw about

½ the current per lamp as the 2-lamp circuit. Since the 1-lamp ballast used a normal power factor design (low PF), the 1-lamp circuit (serving ½ the number of lamp) actually drew more current than 2-lamp circuits. That explained many of the overloads.

Despite these discoveries, the question of harmonics remained. Fluorescent ballasts (even with magnetic ballasts) generate harmonics. This has been common knowledge since at least 1968, when the NEC began to require a full size neutral conductor on circuits serving discharge lighting (which includes fluorescent) and also electronic equipment.

Solution To avoid problems of overloading the neutral on 3phase, 4 wire system,

manufactures of CBM certified ballast limit the 3rd harmonic to 33% of the fundamental so as not to overload the neutral.

Our case involved 2 phases sharing a neutral at each fixture. When phase A and phase B (each with a 30% third harmonic) combine on the neutral. The wave form seriously distorts. The peak to peak is 1.6 time the peak to peak value of the pure sine wave.

A true rms meter would indicate the correct value of 0.82A. However, the electrician’s average responding meter interpreted this current at 1.13A. This measurement error compounded the problems caused by the ballast substitution and the original miscalculation.

The remedy involved new light fixtures

for this space, replacing these fixtures

was less expensive than rewiring. Replaced the fixtures with the 3-lamp

switch-able ballasts and low PF 1-lamp

operation.

Reference

http://www.creativec.com/harmonics.pdf http://www.netaworld.org/files/ItemFileA440.pdf http://www.tvss.net/pq/har.htm http://www.cpccorp.com/harmonic.htm http://leeh.ee.tut.fi/harmo/1.htm http://leeh.ee.tut.fi/harmo/harmoeng.htm

http://www.ecmweb.com/ar/electric_unexpected_harmonics_creates/

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